My grandson is ten and his little sister is three. There are also a 13yo and a 1 yo. The older two "baby" the younger two. There is no competition. Each child participates in activities appropriate to their age but they also do family things together during which all the kids are involved with each other in a variety of ways. In most situations the older ones are not expected to pay attention to the younger ones full time. They spend part of the time just being a kid at their own age. The younger kids are the primary responsibility of the parents; not the older kids unless the agreement ahead of time is that this time older ones will help with younger one.
I wonder if you are trying too hard to get them to be together. Why would you take your 3yo to an event for 9yos? Of course son wants to focus on his friend of the same age. I suggest that expecting him to always include his little sister may be causing him to be less willing to spend time with her.
Perhaps it would help improve their relationship to talk about boundaries; that you expect the older two to help with their little sister some of the time. Talk about specific things. And that other times their time is their own. Then keep an ongoing conversation about how that's working for him.
I'd also set up times during which you play with both of them modeling for both of them ways to have fun together. Be physically active. My grandchildren tickle and roll on the floor. My grandson tells silly jokes. They play catch with a big light ball. They chase each other. But not when my grandson has a friend his own age around. Then one of the adults or the 13 yo takes over with the 3yo. Their mother monitors and separates them before things get out of control, before either gets tired or too silly or too rough.
You mention not wanting to chase the 3yo around the yard. I suggest working on getting more control with her. Perhaps try 1-2-3 Magic. Having to chase her is a good reason to be selective about taking her places.
Maybe I don, t understand what you mean about keeping everyone together. I suggest that with the age difference you cannot keep everyone together all of the time. Each child has different needs attention wise. Do whole family activities but know when it comes to entertainment and recreation they are very different. You do for practical reasons have 2 families.